Numerically, when solving for \(\omega \) , some roots will disappear between inversions simply because they have a smaller basin of attraction in the new inversion. As far as we can tell, however, QNMs numerically appear across multiple inversions, whereas the roots of \(\alpha _n^{(4)}\gamma _{n+1}^{(4)} = 0\) 
only appear for one inversion.
We do not believe these roots are QNMs for the following reasons.
They do not appear in other numerical studies that use different methods, such as [1]}, [2]}, [3]}, [4]}, [5]}, or in analytic studies, such as [6]}, [7]}.
In fact, for each QNM found in previous numerical studies we find an infinite family of spurious roots branching from it.
For the roots on these branches that we have checked, they all satisfy the equation \(\alpha _n^{(4)}\gamma _{n+1}^{(4)} = 0\) .
For the above reasons, we disregard any root that does not persist for at least two inversions.
The one exception is the first inversion, since the roots of Eq. (REF ) are the QNMs.
For example, the purely imaginary root \(\omega = -2i\)  for gravitational waves with \(\ell =2\) , \(r_+ = 1\)  found in [3]}, [4]}, [10]} (but not in [5]}) is only visible in the 1st inversion, where \(\alpha _1^{(4)} = 0\)  and \(\beta _1^{(4)} = 0\) . This \(\omega \)  is not found in higher inversions which are undefined when \(\beta _1^{(4)} = 0\) .
